FACTORING TRINOMIALS, ALL MIXED UP

Here, you will practice factoring trinomials of the form $\,x^2 + bx + c\,$,
where $\,b\,$ and $\,c\,$ are integers.
Notice that the coefficient of the $\,x^2\,$ term is $\,1\,$.

Recall that the integers are:   $\,\ldots,-3,-2,-1,0,1,2,3,\ldots$

As discussed in Basic Concepts Involved in Factoring Trinomials,
you must first find two numbers that add to $\,b\,$ and that multiply to $\,c\,$, since then: $$ \,x^2 + bx + c \ \ =\ \ x^2 + (\overset{=\ b}{\overbrace{f+g}})x + \overset{=\ c}{\overbrace{\ fg\ }} \ \ =\ \ (x + f)(x + g) $$

As discussed in Factoring Trinomials of the form $\,x^2 + bx + c\,$, where $c\gt 0$:
if $\,c\,$ is positive, then both numbers will be positive, or both numbers will be negative.
When you add numbers that have the same sign, then in your head you do an addition problem.

As discussed in Factoring Trinomials of the form $\,x^2 + bx + c\,$, where $c\lt 0$:
if $\,c\,$ is negative, then one number will be positive, and the other will be negative.
When you add numbers that have different signs, then in your head you do a subtraction problem.

the ‘PANS’ memory device

When you're trying to find the two numbers that work,
you always want to do the mental arithmetic with only positive numbers.
It's much easier this way.
Here are the steps:

EXAMPLES:
Factor: $x^2 + 5x + 6$
Solution:
Factor: $x^2 + 5x - 6$
Solution:
Factor: $x^2 - 5x + 6$
Solution:
Factor: $x^2 - 5x - 6$
Solution:
Factor: $x^2 + 3x - 7$
Solution:
Master the ideas from this section
by practicing the exercise at the bottom of this page.

When you're done practicing, move on to:
Identifying Quadratic Equations

 
 

For more advanced students, a graph is displayed.
For example, suppose you're asked to factor $\,x^2 + 5x - 6\,$.
Then, you'll see the graph of $\,y = x^2 + 5x - 6\,$.
Pay attention to where it crosses the $\,x\,$-axis (at $\,-6\,$ and $\,1\,$).
Compare it with the factorization:   $x^2 + 5x - 6 = (x + 6)(x - 1)\,$
See any relationship?
You're discovering the beautiful relationship between the zeroes of a polynomial, and its factors!
Click the “show/hide graph” button if you prefer not to see the graph.

If a trinomial is not factorable over the integers, input ‘NF’ (for ‘Not Factorable’).

Factor: